Contents

Idea

In Link Groups , John Milnor introduced the notion of the Link Group? as a way to study links. The notion of equivalence of links that Milnor used is slightly different to that obtained by extending the usual notion of equivalence of knots. In Milnor’s paper, the crucial aspect of links was the interactions between distinct components. Thus for Milnor, a link in a manifold MM is a map ∐nS1→M\coprod_n S^1 \to M such that the components have disjoint images. Similarly, two links are homotopic if there is a homotopy between the maps which is a link at every time. Thus links can be deformed in such a manner that individual components can pass through themselves, but not through other components. Also link components can have self-intersections or the map on a component can be a constant map. Milnor uses the term proper link to refer to a link in which the map is a homeomorphism onto its image.

The Whitehead link is a simple example of a link that is not trivial under ambient isotopy but is trivial under Milnor’s notion of homotopy.

The μ\mu-invariants come from explicit descriptions of the link groups of particular links. Specifically, Milnor calls a link almost trivial if every proper sublink is trivial (see Brunnian link). Such a link corresponds to an element in a particular link group which can be completely described by certain numbers.

Link Group

Let us begin by describing the link group. Milnor’s alternative description is as follows. Consider the complement of a link LL in an open 33-manifold MM. We choose a basepoint in this complement and so have the fundamental group. We define a relation on this group as follows: two loops α\alpha, β\beta are equivalent if the link L∪α−1βL \cup \alpha^{-1} \beta is homotopic in MM to one of the form L′∪1L' \cup 1 (where 11 is the constant loop at the basepoint). The link group is the group of equivalence classes of such loops.

A more practical description is the following.

Definition

Let LL be a link in an open 33-manifold MM. Let G(L)G(L) be the fundamental group of the complement of LL. Let LiL^i denote the sublink obtained by deleting the iith component of LL. Let Ai(L)A_i(L) be the kernel of the natural inclusion G(L)→G(Li)G(L) \to G(L^i) and [Ai][A_i] its commutator subgroup. Let E(L)=[A1][A2]⋯[An]E(L) = [A_1] [A_2] \cdots [A_n]. This is a normal subgroup of G(L)G(L). The quotient, 𝒢(L)≔G(L)/E(L)\mathcal{G}(L) \coloneqq G(L)/E(L) is the link group of LL.

Milnor’s first theorem on this group was to show that this group is an invariant of the homotopy class of the link, at least for proper links.

If two proper links are homotopic, then their link groups are isomorphic.

To study this group for a particular link, we need to find some particular elements in it. These are the meridians and the parallels. Basically, a meridian goes around one component of the link once, in a specific direction, whilst a parallel goes along it. Technically, the parallels of a link are not elements of its link group, but cosets.

Choose a component LiL_i of the link LL. Choose orientations of the ambient manifold, MM, and of the circle. To define the meridian and parallel of LiL_i we need to choose a path from the basepoint, x0x_0, to a point on the image of LiL_i which does not intersect the image of LL at any other time. Let pp be such a path, so then p(1)p(1) is a point on the image of LiL_i.

Definition

The iith meridian of LL is the element αi∈𝒢(L)\alpha_i \in \mathcal{G}(L) defined as follows. Choose a small neighbourhood NN of p(1)p(1). Define a path by going along pp until we are inside NN, then go around a closed loop in NN which has linking number +1+1 with the part of the image of LiL_i inside NN. Then return to x0x_0 along pp.

The iith parallel of LL is the coset βi𝒜i∈𝒢(Li)\beta_i \mathcal{A}_i \in \mathcal{G}(L^i) defined as follows. The subgroup 𝒜i\mathcal{A}_i is the kernel of the homomorphism 𝒢(L)→𝒢(Li)\mathcal{G}(L) \to \mathcal{G}(L^i). Go along pp to its end. Then go around the image of LiL_i according to the orientation of the circle. Finally return to x0x_0 along pp. The preimage of this element defines a coset in 𝒢(L)\mathcal{G}(L) which we write as βi𝒜i\beta_i \mathcal{A}_i.

The basic method of studying a link via link groups is to consider a link as an element of the link group of the link obtained by removing one of its components. To show that this is a reasonable thing to do, Milnor proved the following theorem.

Let LL be a proper link with nn components. Let ff, f′f' be closed loops in the complement of LL. If they represent conjugate elements of 𝒢(L)\mathcal{G}(L) then the links (L,f)(L,f) and (L,f′)(L,f') are homotopic.

μ\mu-Invariants

For Brunnian links, which Milnor calls almost trivial links, the classification question reduces to looking at elements of the link group of trivial links. It is important to note that the ambient space here is Euclidean space, ℝ3\mathbb{R}^3.

Let LL be an nn-component Brunnian link. Then we consider the element β′n∈𝒢(Ln)\beta'_n \in \mathcal{G}(L^n) corresponding to the nnth parallel. Upon removing a further component, say the n−1n-1st, this element becomes trivial since we are then looking at Ln−1L^{n-1} which is trivial. Thue β′n∈𝒜n−1(Ln)\beta'_n \in \mathcal{A}_{n-1}(L^n), the kernel of 𝒢(Ln)→𝒢(Ln−1,n)\mathcal{G}(L^n) \to \mathcal{G}(L^{n-1,n}) (here Ln−1,nL^{n-1,n} is LL with both the n−1n-1st and nnth components removed). Now 𝒜n−1(Ln)\mathcal{A}_{n-1}(L^n) is the smallest normal subgroup containing the meridian αn−1\alpha_{n-1} (since removing the n−1n-1st component is the same thing as allowing the meridian αn−1\alpha_{n-1} to collapse) and so every element of 𝒜n−1\mathcal{A}_{n-1} can be written as a word in alphabet of powers and conjugates of αn−1\alpha_{n-1}. Milnor uses the notation

In the second identity, the summation is over all shuffle products of (i1⋯iν)(i_1 \cdots i_{\nu}) with (j1⋯jn−ν−2)(j_1 \cdots j_{n - \nu - 2}).

Let us expand on the definition of the μ\mu-invariants. We start with the exponential notation. The following holds for an arbitrary proper link, LL, embedded in an open 33-manifold MM.

Let J𝒢(L)J \mathcal{G}(L) be the integral group ring of 𝒢(L)\mathcal{G}(L). As mentioned above, any element of 𝒜i(L)\mathcal{A}_i(L) is a product of powers of conjugates of αi\alpha_i. We can write such an element in the form αis\alpha_i^s for s∈J𝒢(L)s \in J \mathcal{G}(L) by interpreting:

Now the notation αis\alpha_i^s for an element of 𝒜i(L)\mathcal{A}_i(L) does not provide an injective map from J𝒢(L)J\mathcal{G}(L) to 𝒜i(L)\mathcal{A}_i(L). The kernel is the ideal 𝒦i(L)+(𝒦1(L)2+⋯+𝒦n(L)2)\mathcal{K}_i(L) + (\mathcal{K}_1(L)^2 + \cdots + \mathcal{K}_n(L)^2) which is naturally isomorphic to ℛ(Li)\mathcal{R}(L^i).

Let LL be a link which is homotopic to one in with the iith component is constant. Then every element of 𝒜i(L)\mathcal{A}_i(L) can be expressed uniquely in the form αiσ\alpha_i^\sigma with σ∈ℛ(Li)\sigma \in \mathcal{R}(L^i).

Now let us suppose that LL is trivial. Then G(L)G(L) is the free product of the fundamental group of MM with the infinite cyclic groups generated by the (elements representing the) meridians of LL. Let these be a1a_1, …, ana_n and let ki=ai−1k_i = a_i - 1 in JG(L)J G(L). Milnor defines a canonical word to be a product of the form ϕ0kj1ϕ1kj2ϕ2⋯kjpϕp\phi_0 k_{j_1} \phi_1 k_{j_2} \phi_2 \cdots k_{j_p} \phi_p with p≥0p \ge 0, ϕi∈π1(M)\phi_i \in \pi_1(M), and 1≤ji≤n1 \le j_i \le n. A canonical sentence is a sum or difference of any number of canonical words. It turns out (Milnor, Theorem 7) that each element of ℛ(L)\mathcal{R}(L) is represented by a unique canonical sentence.

Now let us return to the case of the almost trivial link in Euclidean space. From above, we have the element β′n∈𝒜i(Ln)\beta'_n \in \mathcal{A}_i(L^n) corresponding to the nnth parallel. Removing any other component allows us to trivialise β′n\beta'_n since removing, say, the iith component leaves us with LiL^i which is homotopic to the trivial link on n−1n-1 components. Removing the ii component corresponds to setting aia_i to 11 in JG(Ln)J G(L^n), equivalently to setting ki=0k_i = 0. So upon setting ki=0k_i = 0 we must have that β′n↦1\beta'_n \mapsto 1 and thus (by uniqueness) σ↦0\sigma \mapsto 0. Hence kik_i divides σ\sigma, and so every canonical word in σ\sigma is of the form ki1⋯kin−2k_{i_1} \cdots k_{i_{n-2}} for some permutation of 11, 22, …, n−2n-2. Sorting them out by permutation, we get the expression in (1).

Now, how do we interpret or calculate these invariants? We need to work out what an expression of the form in (1) is saying. Consider a canonical word, ki1⋯kin−2k_{i_1} \cdots k_{i_{n-2}}. The corresponding element is:

αn−1ki1⋯kin−2
\alpha_{n-1}^{k_{i_1} \cdots k_{i_{n-2}}}

Let us write α=αn−1\alpha = \alpha_{n-1}. Now αk1\alpha^{k_1} is αa1−1=a1αa1−1α−1\alpha^{a_1 - 1} = a_1 \alpha a_1^{-1} \alpha^{-1}. Thus this tells us to go around L1L_1, then Ln−1L_{n-1}, back around L1L_1, and finally back around Ln−1L_{n-1}. Each time we introduce a new power, we do the same except that we replace the loop around Ln−1L_{n-1} with the loop so far constructed.

So the general method is as follows: choose two components of the link. Write one of them as a word in the meridians of the others. Then simplify this word using the other chosen link as the “base”: namely, write everything in terms of conjugates of that base. This will then separate out into the desired form and, hopefully, the link invariants can be read off.